Vanishing and cocentralizing generalized derivations on Lie ideals

Carini, Luisa; Filippis, Vincenzo De; Scudo, Giovanni

doi:10.1080/00927872.2018.1439043

Cited by 2 publications

(3 citation statements)

References 19 publications

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“…By Proposition 16 in [4], we conclude that either a, b ∈ C or q − u ∈ C, a contradiction. Hence there is no conclusion of Proposition 3.5.…”

Section: The Case: Inner Generalized Derivationsmentioning

confidence: 83%

“…The center C = Z(U) is called the extended centroid of R. By d, we mean a nonzero derivation of R. For x, y ∈ R, [x, y] = xy − yx is the commutator of x and y. The s 4 denotes the standard polynomial in four variables, which is s 4 (x 1 , x 2 , x 3 , x 4 ) = σ∈S 4 (−1) σ x σ (1) x σ (2) x σ (3) x σ (4) , where (−1) σ is +1 or −1 according to σ being an even or odd permutation in symmetric group S 4 . Let S be a nonempty subset of R. An additive mapping f : R → R is said to be commuting (respectively, centralizing) on S, if [ f (x), x] = 0 for all x ∈ S (respectively, [ f (x), x] ∈ Z(R) for all x ∈ S).…”

Section: Introductionmentioning

confidence: 99%

“…Note that recently Carini et al [4] studied the case F(G(u)u − uH(u)) = 0 for all u in a noncentral Lie ideal in prime ring R.…”

Section: Introductionmentioning

confidence: 99%

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Generalized derivations vanishing on co-commutator identities in prime rings

Dhara¹

2021

Filomat

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Let R be a noncommutative prime ring of char (R)? 2 with Utumi quotient ring U and extended centroid C and I a nonzero two sided ideal of R. Suppose that F(? 0), G and H are three generalized derivations of R and f (x1,...,xn) is a multilinear polynomial over C, which is not central valued on R. If F(G(f(r))f(r)- f(r)H(f(r))) = 0 for all r = (r1,..., rn) ? In, then we obtain information about the structure of R and describe the all possible forms of the maps F, G and H. This result generalizes many known results recently proved by several authors ([1], [4], [5], [8], [9], [13], [15]).

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“…By Proposition 16 in [4], we conclude that either a, b ∈ C or q − u ∈ C, a contradiction. Hence there is no conclusion of Proposition 3.5.…”

Section: The Case: Inner Generalized Derivationsmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

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Generalized derivations vanishing on co-commutator identities in prime rings

Dhara¹

2021

Filomat

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An Engel condition with two generalized derivations on Lie ideals of prime rings

Liu

2022

J. Algebra Appl.

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Let [Formula: see text] be a prime ring, let [Formula: see text] be a noncentral Lie ideal of [Formula: see text] and let [Formula: see text] be two generalized derivations of [Formula: see text]. In this paper, we characterize the structure of [Formula: see text] and all possible forms of [Formula: see text] and [Formula: see text] such that [Formula: see text] for all [Formula: see text], where [Formula: see text] are fixed positive integers. With this, several known results can be either deduced or generalized. In particular, we give a Lie ideal version of the theorem obtained by Lee and Zhou in [An identity with generalized derivations, J. Algebra Appl. 8 (2009) 307–317] and describe a more complete version of the theorem recently obtained by Dhara and De Filippis in [Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings, Comm. Algebra 48 (2020) 154–167].

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Vanishing and cocentralizing generalized derivations on Lie ideals

Cited by 2 publications

References 19 publications

Generalized derivations vanishing on co-commutator identities in prime rings

Generalized derivations vanishing on co-commutator identities in prime rings

An Engel condition with two generalized derivations on Lie ideals of prime rings

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